Power set
The set of all subsets of a given set.
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In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The power set is a fundamental concept in set theory, with applications in Boolean algebra, measure theory, and combinatorics.
Quick Facts
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Facts from the source article.
Background
The power set of a set S is defined as the collection of all subsets of S, including the empty set and S itself. For a finite set with n elements, the power set contains exactly 2^n subsets, a fact that motivates the notation 2^S. This equivalence is demonstrated through indicator functions: each subset corresponds to a function from S to {0,1}, and the set of all such functions is denoted {0,1}^S, which is bijective to the power set.
Frequently Asked Questions
How do you write or denote the Power set?
You will see it rendered as P(S), π«(S), β(S), or even 2^S, with the script P being the most common in formal texts. The 2^S notation is a hint at the exponential relationship between the size of S and the size of its power set.
If a set has n elements, how many subsets does its Power set contain?
The power set of an n-element set always has exactly 2^n elements. For example, a three-element set like {x, y, z} yields a power set of eight distinct subsets.
Why is Cantor's diagonal argument tied to the Power set?
Cantor's diagonal argument proves that no set can be placed in one-to-one correspondence with its own power set, meaning the power set always has strictly greater cardinality. This result is the engine behind the fact that there is no single 'largest' infinity.
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Sources
Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.
- Wikipedia: Power set (CC BY-SA 4.0).
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